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Wolfgang Lück

Publications and source records attributed to Wolfgang Lück.

At least 19 recordsLinked to original sources
arXiv2024

Character theory and Euler characteristic for orbispaces and infinite groups

Given a discrete group $G$ with a finite model for $\underline{E}G$, we study $K(n)^*(BG)$ and $E^*(BG)$, where $K(n)$ is the $n$-th Morava $K$-theory for a given prime and $E$ is the height $n$ Morava $E$-theory. In particular we generalize the character theory of Hopkins, Kuhn and Ravenel who studied these objects for finite groups. We give a formula for a localization of $E^*(BG)$ and the $K(n)$-theoretic Euler characteristic of $BG$ in terms of centralizers. In certain cases these calculations lead to a full computation of $E^*(BG)$, for example when $G$ is a right angled Coxeter group, and for $G=SL_3(\mathbb{Z})$. We apply our results to the mapping class group $Γ_\frac{p-1}{2}$ for an odd prime $p$ and to certain arithmetic groups, including the symplectic group $Sp_{p-1}(\mathbb{Z})$ for an odd prime $p$ and $SL_2(\mathcal{O}_K)$ for a totally real field $K$.

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arXiv2021

Equivariant Cohomology of Configuration Spaces mod 2: The State of the Art

The equivariant cohomology of the classical configuration space $F(\mathbb{R}^d,n)$ has been been of great interest and has been studied intensively starting with the classical papers by Artin (1925/1947) on the theory of braids, by Fox and Neuwirth (1962), Fadell and Neuwirth (1962), and Arnol'd (1969). We give a brief treatment of the subject from the beginnings to recent developments. However, we focus on the mod 2 equivariant cohomology algebras of the classical configuration space $F(\mathbb{R}^d,n)$, as described in an influential paper by Hung (1990). We show with a new, detailed proof that his main result is correct, but that the arguments that were given by Hung on the way to his result are not, as are some of the intermediate results in his paper. This invalidates a paper by three of the present authors, Blagojević, Lück \& Ziegler (2016), who used a claimed intermediate result from Hung (1990) in order to derive lower bounds for the existence of $k$-regular and $\ell$-skew embeddings. Using our new proof for Hung's main result, we get new lower bounds for existence of highly regular embeddings: Some of them agree with the previously claimed bounds, some are weaker.

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arXiv2020

Proper equivariant stable homotopy theory

This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective `proper' alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from equivariant cells with compact isotropy groups; the adjective `genuine' indicates that the theory comes with appropriate transfers and Wirthmüller isomorphisms, and the resulting equivariant cohomology theories support the analog of an $RO(G)$-grading. Our model for genuine proper $G$-equivariant stable homotopy theory is the category of orthogonal $G$-spectra; the equivalences are those morphisms that induce isomorphisms of equivariant stable homotopy groups for all compact subgroups of $G$. This class of $π_*$-isomorphisms is part of a symmetric monoidal stable model structure and the associated tensor triangulated homotopy category is compactly generated. Every orthogonal $G$-spectrum represents an equivariant cohomology theory on the category of $G$-spaces, depending only on the `proper $G$-homotopy type', tested by fixed points under all compact subgroups. An important special case are infinite discrete groups. For these, our genuine equivariant theory is related to finiteness properties, in the sense of geometric group theory; for example, the $G$-sphere spectrum is a compact object in the equivariant homotopy category if the universal space for proper $G$-actions has a finite $G$-CW-model. For discrete groups, the represented equivariant cohomology theories on finite proper $G$-CW-complexes admit a more explicit description in terms of parameterized equivariant homotopy theory, suitably stabilized by $G$-vector bundles. Via this description, we can identify the previously defined $G$-cohomology theories of equivariant stable cohomotopy and equivariant K-theory as cohomology theories represented by specific orthogonal $G$-spectra.

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arXiv2020

A vanishing theorem for tautological classes of aspherical manifolds

Tautological classes, or generalised Miller-Morita-Mumford classes, are basic characteristic classes of smooth fibre bundles, and have recently been used to describe the rational cohomology of classifying spaces of diffeomorphism groups for several types of manifolds. We show that rationally tautological classes depend only on the underlying topological block bundle, and use this to prove the vanishing of tautological classes for many bundles with fibre an aspherical manifold.

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arXiv2019

K- and L-theory of graph products of groups

We compute the group homology, the algebraic $K$- and $L$-groups, and the topological $K$-groups of right-angled Artin groups, right-angled Coxeter groups, and more generally, graph products.

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arXiv2018

$L^2$-Euler characteristics and the Thurston norm

We assign to a finite $CW$-complex and an element in its first cohomology group a twisted version of the $L^2$-Euler characteristic and study its main properties. In the case of an irreducible orientable $3$-manifold with empty or toroidal boundary and infinite fundamental group we identify it with the Thurston norm. We will use the $L^2$-Euler characteristic to address the problem whether the existence of a map inducing an epimorphism on fundamental groups implies an inequality of the Thurston norms.

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arXiv2017

Universal $L^2$-torsion, polytopes and applications to $3$-manifolds

Given an $L^2$-acyclic connected finite $CW$-complex, we define its universal $L^2$-torsion in terms of the chain complex of its universal covering. It takes values in the weak Whitehead group $\operatorname{Wh}^w(G)$. We study its main properties such as homotopy invariance, sum formula, product formula and Poincaré duality. Under certain assumptions, we can specify certain homomorphisms from the weak Whitehead group $\operatorname{Wh}^w(G)$ to abelian groups such as the real numbers or the Grothendieck group of integral polytopes, and the image of the universal $L^2$-torsion can be identified with many invariants such as the $L^2$-torsion, the $L^2$-torsion function, twisted $L^2$-Euler characteristics and, in the case of a $3$-manifold, the dual Thurston norm polytope.

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arXiv2016

Localization, Whitehead groups, and the Atiyah Conjecture

Let Wh^w(G) be the K_1-group of square matrices over the integral group ring ZG which are not necessarily invertible but induce weak isomorphisms after passing to Hilbert space completions. Let D(G) be the division closure of ZG in the algebra U(G) of operators affiliated to the group von Neumann algebra. Let C be the smallest class of groups which contains all free groups and is closed under directed unions and extensions with elementary amenable quotients. Let G be a torsionfree group which belongs to C. Then we prove that Wh^w(G) is isomorphic to K_1(D(G)). Furthermore we show that D(G) is a skew field and henc K_1(\D(G)) is the abelianization of the multiplicative group of units in D(G).

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arXiv2016

On the Farrell-Jones Conjecture for Waldhausen's $A$-theory

We prove the Farrell-Jones Conjecture for (non-connective) $A$-theory with coefficients and finite wreath products for hyperbolic groups, CAT(0)-groups, cocompact lattices in almost connected Lie groups and fundamental groups of manifolds of dimension less or equal to three. Moreover, we prove inheritance properties such as passing to subgroups, colimits of direct systems of groups, finite direct products and finite free products. These results hold also for Whitehead spectra and spectra of stable pseudo-isotopies in the topological, piecewise linear and smooth category.

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arXiv2016

Groups and polytopes

In a series of papers the authors associated to an $L^2$-acyclic group $Γ$ an invariant $\mathcal{P}(Γ)$ that is a formal difference of polytopes in the vector space $H_1(Γ;\Bbb{R})$. This invariant is in particular defined for most 3-manifold groups, for most 2-generator 1-relator groups and for all free-by-cyclic groups. In most of the above cases the invariant can be viewed as an actual polytope. In this survey paper we will recall the definition of the polytope invariant $\mathcal{P}(Γ)$ and we state some of the main properties. We conclude with a list of open problems.

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arXiv2015

The L^2-Alexander torsion of 3-manifolds

We introduce $L^2$-Alexander torsions for 3-manifolds, which can be viewed as a generalization of the $L^2$-Alexander polynomial of Li--Zhang. We state the $L^2$-Alexander torsions for graph manifolds and we partially compute them for fibered manifolds. We furthermore show that given any irreducible 3-manifold there exists a coefficient system such that the corresponding $L^2$-torsion detects the Thurston norm.

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arXiv2015

The L^2-torsion function and the Thurston norm of 3-manifolds

Let M be an oriented irreducible 3-manifold with infinite fundamental group and empty or toroidal boundary. Consider any element ϕin the first cohomology of M with integral coefficients. Then one can define the ϕ-twisted L^2-torsion function of the universal covering which is a function from the set of positive real numbers to the set of real numbers. By earlier work of the second author and Schick the evaluation at t=1 determines the volume. In this paper we show that its degree, which is a number extracted from its asymptotic behavior at 0 and at infinity, agrees with the Thurston norm of ϕ.

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arXiv2015

On complex highly regular embeddings and the extended Vassiliev conjecture

A continuous map C^d -> C^N is a complex k-regular embedding if any k pairwise distinct points in C^d are mapped by f into k complex linearly independent vectors in C^N. Our central result on complex k-regular embeddings extends results of Cohen & Handel (1978), Chisholm (1979) and Blagojevic, Lück & Ziegler (2013) on real k-regular embeddings: We give new lower bounds for the existence of complex k-regular embeddings. These are obtained by modifying the framework of Cohen & Handel (1978) and a study of Chern classes of complex regular representations. The main technical result, used for the study of the Chern classes, is an upper bound for the height of the cohomology of an unordered configuration space Furthermore, we give similar lower bounds for the existence of complex l-skew embeddings C^d -> C^N, for which we require that the images of the tangent spaces at any l distinct points are skew complex affine subspaces of C^N.

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arXiv2014

The $L^2$-Alexander torsion is symmetric

We show that the $L^2$-Alexander torsion of a 3-manifold is symmetric. This can be viewed as a generalization of the symmetry of the Alexander polynomial of a knot.

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arXiv2014

Three flavors of twisted invariants of knots

The Alexander polynomial of a knot has been generalized in three different ways to give twisted invariants. The resulting invariants are usually referred to as twisted Alexander polynomials, higher-order Alexander polynomials and $L^2$-Alexander invariants of knots. We quickly recall the definitions and we summarize and compare some of their properties. We also report on work by the authors on $L^2$-Alexander torsions and we conclude the paper with several conjectures on $L^2$-Alexander torsions.

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arXiv2014

Dwyer-Kan homotopy theory of enriched categories

We construct a model structure on the category of small categories enriched over a combinatorial closed symmetric monoidal model category satisfying the monoid axiom. Weak equivalences are Dwyer-Kan equivalences, i.e. enriched functors which induce weak equivalences on morphism objects and equivalences of ordinary categories when we take sets of connected components on morphism objects.

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arXiv2013

On the growth of Betti numbers in $p$-adic analytic towers

We study the asymptotic growth of Betti numbers in tower of finite covers and provide simple proofs of approximation results, which were previously obtained by Calegari-Emerton, in the generality of arbitrary p-adic analytic towers of covers. Further, we also obtain partial results about arbitrary pro-$p$ towers.

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